Do they sing this at Russian children's birthday parties?

Ответ - нет. Завтра мое рождение means that you have not born yet.

Looking at the hint, it just means birth. I imagine you'd get the same sort of funny looks as if you said "My birth is tomorrow" in English.

Apparently the second half of рождение doesn't come from день so it's not as redundant as it sounds.

Its has a noun suffix added to modify рождать, the imperfective of "to birth" (which may or may not have the idea of "give" included since рожать is an alternate imperfective of родить).

I'm quite sure that "It's my birthday tomorrow." should also be accepted.